Methods and apparatus for coherent duobinary shaped PM-QPSK signal processing

ABSTRACT

Systems, devices and techniques for receiving a signal comprising a quadrature duobinary modulated signal include performing channel equalization of the received signal using a constant multi-modulus to obtain a set of channel estimation coefficients and a stream of symbols, partitioning, based on modulus, the stream of symbols into three partitions, estimating carrier frequency based on the partitioned stream of symbols, recovering a phase of the signal using a maximum likelihood algorithm, and decoding the partitioned stream of symbols to recover data.

CROSS REFERENCE TO RELATED APPLICATIONS

This patent application claims the benefit of U.S. Provisional PatentApplication No. 61/681,462, filed on Aug. 9, 2012. The entire content ofthe before-mentioned patent application is incorporated by reference aspart of the disclosure of this application.

TECHNICAL FIELD

This patent document relates to systems, devices and techniques forprocessing of optical signals.

BACKGROUND

Networks that use optical communications medium, such as fiber optic,are becoming increasingly popular to meet ever-growing bandwidth demand.Optical networks are often used to carry high bandwidth video datato/from users to the network and also in backhaul operation of anetwork.

Techniques are needed for improved optical communications performance.

SUMMARY

This patent document provides, among others, systems, devices andtechniques that are useful in improving the performance of opticalcommunications receivers.

In one aspect, the disclosed techniques include a blind polarizationde-multiplexing technique based on a cascaded multi-modulus algorithm, apartitioning of quadrature phase signal keying (QPSK) usingmulti-modulus frequency offset estimation (FOE), and carrier phaserecovery (CPR) with maximum likelihood (ML) phase estimation.

This and other aspects and their implementations are described ingreater detail in the drawings, the description and the claims.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows (a) example Quadrature Duo-Binary (QDB) spectrum shapedPolarization Modulated QPSK (PM-QPSK) by WSS or waveshaper for one pol.;(b) Constellation of PM-QPSK signal before and after QDB spectrumshaping.

FIG. 2 illustrates CMMA for QDB spectrum shaped PDM-QPSK.

FIG. 3 illustrates QPSK partition and rotation.

FIG. 4 is a block diagram representation of joint-polarization QPSKpartitioning FOE.

FIG. 5 is a block diagram representation of two stages phase estimationbased on QPSK partitioning/ML.

FIG. 6 illustrates an example in which normalized taps amplitudefrequency response for CMMA and CMA under different spectrum shapingbandwidth.

FIGS. 7A-7B illustrate (A) Impact of block size N on the performance ofFOE; (B) performance of FOE algorithm for different frequency offset.

FIGS. 8A-8B are a graphical representation of (A) Optimum group size mvarying with OSNR for different linewidth and QDB bandwidth; and (B)OSNR penalty at BER of 1E-3 varying with linewidth for different QDBbandwidth.

FIG. 9 shows simulation results of BER performance varying with QDBspectrum shaping bandwidth for different DSP schemes.

FIG. 10 illustrates back to back BER performance varying with OSNR fordifferent DSP schemes.

FIG. 11 is a flowchart representation of a process of opticalcommunication.

FIG. 12 is a block diagram representation of a portion of an opticalcommunications receiver.

FIG. 13 depicts a block diagram representation of an opticalcommunication system.

DETAILED DESCRIPTION

A novel digital signal processing scheme (DSP) for quadrature duobinary(QDB) spectrum shaped polarization multiplexed quadrature phase shiftkeying (PM-QPSK) based on multi-modulus blind equalizations (MMBE) isdisclosed and demonstrated with both simulation and experimentalresults. The algorithms for this novel digital signal processing schemeinclude the cascaded multi-modulus algorithm (CMMA) for blindpolarization de-multiplexing, multi-modulus QPSK partitioning frequencyoffset estimation (FOE) and two stage carrier phase recovery (CPR) withmaximum likelihood phase estimation. The final signal is detected bymaximum-likelihood sequence detection (MLSD) for data BER measurement.The feasibility of the disclosed digital signal processing scheme isdemonstrated by the experiment of 112 Gb/s QDB spectrum shaped PM-QPSKsignal with a 25 GHz bandwidth waveshaper for Nyquist WDM channels.

The following abbreviations are used in the present document.

Acronym Fullform ADC Analog to Digital Conversion BER Bit Error Rate CDChromatic Dispersion CMA Constant Modulus Algorithm CMBE ConstantModulus Blind Equalization CMMA Cascaded Multi Modulus Algorithm CPRCarrier Phase Recovery DSP Digital Signal Processing/Processor ECLExternal Cavity Laser FOE Frequency Offset Estimation ML MaximumLikelihood MLSD Maximum Likelihood Sequence Detection MMBE Multi ModulusBlind Equalization NWDM Nyquist Wave Division Multiplexing OC OpticalCarrier OSNR Optical Signal To Noise Ratio PBC Polarization BeamCombiner PDM Polarization Division Multiplexing PM PolarizationModulation PPG Pulse Pattern Generator QAM Quadrature AmplitudeModulation QDB Quadrature Duo Binary QPSK Quadrature Phase Shift KeyingRD-CMA Radius Directed Constant Modulus Algorithm SC Single Carrier SESpectral Efficiency VVPE Viterbi - Viterbi Phase Estimation WSSWavelength Selective Switch

FIG. 13 is a block diagram representation of an optical communicationsystem 100 where the subject technology disclosed of this document canbe implemented. An optical transmitter 102 transmits optical signalsthrough an optical network 104 to one or more optical transceivers 106.The transmitted optical signals may go through intermediate opticalequipment such as amplifiers, repeaters, switch, etc., which are notshown in FIG. 13 for clarity. The disclosed transmission techniques canbe implemented in the transmission subsystem of the transmitter 102. Thedisclosed reception techniques can be implemented in the receiversubsystem of the receiver 106.

Recently, QDB spectrum shaping technique has attracted a lot ofattentions due to its nearly doubled SE and tolerance to channelcrosstalk and CD with respect to QPSK signals especially in 100 G (100Gigabit per second) and 200 G (200 Gigabit per second) coherent opticalcommunications. A SE of >4 bit/s/Hz has been demonstrated by using theQDB format and PolMux scheme with enhanced tolerance to the narrowoptical filtering. However, due to constellation zero point caused byfiltering effect, the conventional CMBE algorithms for PM QPSK coherentdetection are not compatible with the new techniques. To allow for useof conventional DSP-based schemes, some conventional techniques usepre-filtering and post-filtering stages. Eighth power based Viterbicarrier phase recovery (CPR) has also been used in QDB systems. However,only one polarization is considered in the conventional techniques. A PMQDB system with radius-directed constant modulus algorithm (RD-CMA) hasalso been previously proposed. However, this implementation fails totake into account some key adaptive equalization including carrierfrequency offset estimation and phase recovery. In conventional art, acascaded multi-modulus algorithm (CMMA) algorithm used in PM 8-QAMsystems shows good modulus decision performances, which gives apossibility for multi-modulus blind equalizations (MMBE) used in QDBPM-QPSK system. On the other hand some conventional embodiments show theadvantages of maximum-likelihood sequence detection (MLSD) for QDBPM-QPSK signals.

In this document, we disclose and provide results of experiments for adigital signal processing (DSP) scheme for QDB spectrum shaped PM-QPSKbased on MMBE. In some embodiments, a CMMA algorithm for blindpolarization de-multiplexing, multi-modulus QPSK partitioning FOE andtwo stages CPR with ML phase estimation is disclosed. The final signalis detected by MLSD for data BER measurement. The feasibility of thedisclosed digital signal processing scheme is demonstrated by theexperiment of 112 Gb/s QDB spectrum shaped PM-QPSK signal with a 25 GHzbandwidth waveshaper for Nyquist WDM (NWDM) channels.

With reference to FIG. 1(a) and FIG. 1(b), the spectral shaping can beperformed by either two electrical low-pass filters in electrical domainor an optical band-pass filter after the optical QPSK modulation inoptical domain with same performance. For PM-QPSK signals with symbolrate of Rs, we use a waveshaper or wavelength selective switch (WSS) 160with 3 dB pass bandwidth of Rs or less for spectral shaping as shown inFIG. 1 (a). The system 150 includes optical source ECL 152, on which Idata 154 and Q data 156 is modulated to produce I/Q data 158, which isthen passed through the WSS 160. The PM-QPSK signal constellationsbefore and after QDB spectrum shaping are shown in FIG. 1 (b). After QDBspectrum shaping, the 4-point QPSK signal (162) becomes a 9-pointduobinary QPSK signal with zero point in constellation due to thefiltering effect (164). The QDB shaped signal is significantly spectrumnarrower compared with QPSK, and the spectral side lobes are alsogreatly suppressed. The 9 points of QDB QPSK signals are located onthree circles with different radii (see FIG. 2).

According the three modulus constellation location, new multi-modulusDSP schemes are disclosed for QDB spectrum shaped signals. These DSPalgorithms including polarization demultiplexing, frequency offsetestimation (FOE) and carrier phase recovery (CPR) are described and thensimulated with 112 Gb/s QDB spectrum shaped PM-QPSK simulation resultsas follows.

Cascaded Multi-Modulus Algorithm for QDB PM-QPSK

For QDB spectrum shaped PM-QPSK, classic CMA is not well compatible.This is because 9-point signal does not present constant symbolamplitude. It not only leads to extra noise after equalization, but alsocauses a problem with filter taps frequency response. Thus, we use CMMAdisclosed and used in PM 8-QAM systems with good modulus decisionperformances for blind polarization de-multiplexing.

Some aspects of CMMA for QDB spectrum shaped PDM-QPSK signals are shownin FIG. 2. It is also a four butterfly-configured adaptive digitalequalizer. Here, ε_(x,y) is the feedback signal error for filter tapupdating. By introducing three reference circles A1˜A3, the final errorcan approach zero for ideal QDB signal as worked in 8 QAM signals. R1˜R3are the radius of the three modulus QDB PDM-QPSK signal and Zx,y is theoutput of equalizer. As a result, it is clear that the regular CMA errorsignal will not approach zero even for an ideal 9-point signal.

Joint-Polarization QPSK Partitioning FOE

The partition scheme has been presented in for FOE in a 16-QAM coherentsystem, the regular m-power algorithm can be also used for FOE for the9-point QDB spectrum shaped signal with partitioning. On the other hand,for polarization multiplexed coherent system, the same transmitter andLO are used for the two polarizations signals. In this way, bothpolarizations signals are affected by the same frequency offset. Toaddress this issue, we use a joint-polarization QPSK partitioningalgorithm for FOE.

FIGS. 3 and 4 show the principle and block diagram 400 for disclosedJoint Polarized QPSK modulation Partition Algorithm (JPMPA) for FOE.After polarization demultiplexing by CMMA, the incoming X and Y pol.symbols (402) are first partitioned into three groups with differentcircle radius (404, 302). In some embodiments, only pairs of consecutiveR1 and R2 symbols are selected (406) for estimation to reduce thecomplexity of the algorithm. Then R2 symbols are first rotated with −π/4angle (408, 304) and then normalized (306). However, R1 symbols are onlynormalized (410). After that, the two groups can combine together with a“QPSK” like constellation. In this way, the 4 power frequency estimationfor QPSK can operate now. For N pairs of R1 and R2 symbols, the phaseangle estimation caused by frequency offset is (412):

$\begin{matrix}{{\Delta\theta}_{est} = {{2{\pi\Delta}\; f_{est}T_{s}} = {\frac{1}{4}\arg{\sum\limits_{1}^{N}\left( {S_{k + 1} \cdot S_{k}^{*}} \right)^{4}}}}} & (1)\end{matrix}$Here, S_(k) is the combined normalized symbols of R1 and R2 groups,T_(s) is the symbol duration and Δf_(est) is the estimated frequencyoffset. Then the frequency offset can be compensated by e^(−jnΔθ) ^(est)for both polarization symbols. The frequency offsetΔf_(est) can beestimated with in [−1/(8 T_(s)), +1/(8 T_(s))] for 4^(th) poweroperation.

Two Stages QPSK Partitioning/ML Carrier Phase Recovery

As analyzed above, the QPSK partitioning scheme can be also used in theCPR for QDB spectrum shaped signals. On the other hand, maximumlikelihood algorithm shows good improvement and low complexity for 16QAM phase estimation. In this way, we propose a two stage phase recoverybased on QPSK Partition/ML is shown in system 500 of FIG. 5 with respectto X/Y polarized input 502.

Some aspects of R1 and R2 ring partition (504), rotation andnormalization (508 and 510) is the same as shown in FIG. 3. In practice,the partition steps for FOE and phase recovery can become one. Afterthat, the symbols in R2 are first rotated with −π/4 angle and thennormalized, while R1 symbols are only normalized and then combined withR2 symbols. Here, all the combined symbols can be used in groups and mis the number of symbols in each group. In this way, the first stagephase φ_(est) ¹ can be eliminated by Viterbi and Viterbi phaseestimation (VVPE) 512 as:

$\begin{matrix}{\phi_{est}^{1} = {\left( {\sum\limits_{m}S_{k}^{4}} \right)/4}} & (2)\end{matrix}$the second stage of phase φ_(est) ² estimation (514) based on ML as:

$\begin{matrix}{{\phi_{est}^{2} = {\tan^{- 1}\left( {{{Im}\lbrack h\rbrack}/{{Re}\lbrack h\rbrack}} \right)}},{h = {\sum\limits_{m}{x_{k} \cdot y_{k}^{*}}}}} & (3)\end{matrix}$Here, yk is the decision of xk after the first stage phase recovery. Thesecond stage phase recovery is implemented before final output.

Performances with Simulation Results

The simulation is carriers out with the disclosed DSP scheme with 112Gb/s PM-QPSK signals. The QDB spectrum shaping is operated by a 4thorder Gaussian optical band pass filter with different 3-dB bandwidthwhich is close to a commercial waveshaper. After DSP mentioned above,the final output is detected by MLSD for data BER measurement.

FIG. 6 shows the normalized taps amplitude frequency response with inNyquist bandwidth for CMMA and CMA taps from 20 GHz to 28 GHz spectrumshaping. Here Rs is the symbol rate. It shows that CMMA have betterperformance with frequency response to compress the noise. For shaperspectrum shaping, the noise around ±Rs/2 can be significantly enhancedfor Nyquist WDM channels. However, CMMA taps are in compression at±Rs/2. The constellations after CMMA are shown as an insertion 602 inFIG. 6.

FIG. 7(a) shows the impact of block size N, in Eq. 1 of the disclosedFOE algorithm on BER performance for different QDB spectrum shapingbandwidth and OSNR (chart 702). Here, we keep the offset frequency withΔf. Ts=0.1 and linewidth of signal source and LO with Δv=100 kHz. We cansee that, N=10000 (near 704) is the optimal block size for FOE underdifferent QDB spectrum shaping bandwidth and OSNR. FIG. 7(b) chart 752shows the performance of FOE results for different frequency offset inthe whole FOE range under different QDB spectrum shaping bandwidth withOSNR at 16 dB. The disclosed FEO algorithm shows good estimationaccuracy within the whole estimation range for different QDB spectrumshaping bandwidth

FIG. 8(a) shows the optimum group size m in Eq. 2 and 3 of disclosed twostage CPR algorithm varies with OSNR for different linewidth and QDBspectrum shaping bandwidth. It shows that, the optimum m deceases withOSNR and linewidth. The optimal m becomes larger and more sensitive toASE noise when they are dominant with smaller linewidth. On the otherhand, when ASE noise gets lager with smaller OSNR, larger group size mis needed for phase estimation and recovery. Simulation results of OSNRpenalty (at BER of 1×10−3) for different QDB spectrum shaping bandwidthvarying with linewidth Δv*Ts is shown in FIG. 8(b). The QDB bandwidth of25 GHz shows the best performance and it can for can tolerate Δv*Ts of5E-4 with OSNR penalty of 1 dB.

FIG. 9 shows the simulation results of back to back BER performancevarying with QDB spectrum shaping bandwidth for the conventionalconstant modulus algorithms of CMBE (906, 908), and our disclosedmulti-modulus algorithms of MMBE for both single carrier (SC) and NWDM.Here we keep the OSNR at 16 dB. It shows that, the disclosed MMBE scheme(902, 904) has better tolerance for strong QDB spectrum shaping and alsocrosstalk form other channels. The conventional CMBE for NWDM has theworst performance due to the crosstalk and strong shaping. It also showsthat the optimal QDB spectrum shaping bandwidth for our disclosed MMBEscheme is from 23 to 25 GHz for SC and 21 to 23 for NWDM

Experiment Results

The effectiveness of the disclosed method has also been tested in a 28Gbaud QDB spectrum shaped NWDM PM-QPSK back-to-back experiment. The NWDMsubchannels are from a comb generator based on phase and intensitymodulators with 25 GHz carrier spacing and equal tone power. For QPSKmodulation, the 28-Gbaud binary electrical signals are generated from anelectrical two channels pulse pattern generator (PPG) with a word lengthof 213-1. The I/Q modulator is biased at the null point and driven atfull swing to achieve zero-chirp 0 and π phase modulation. Thepolarization multiplexing of the signal is realized by the polarizationmultiplexer, which comprises a PM-OC to halve the signal, an opticaldelay line to provide a delay of 150 symbols, and a polarization beamcombiner (PBC) to recombine the signal. The even and odd channels aremodulated and polarization multiplexed individually. After that, theyare combined and QDB spectrum shaped by a commercial WSS with a 3 dBbandwidth of 19.5 GHz and 25 GHz spacing. At the receiver, one tunableband-pass filter (BPF) with 3 dB bandwidth of 0.4 nm is employed tochoose the measured subchannel. Polarization and phase diverse coherenthomodyne detection is employed at the receiver. Here, the linewidth ofECL at the transmitter and LO at the receiver is both smaller than 100kHz. The Analog/Digital conversion (ADC) is realized in the digitalscope with the sample rate of 50-GSa/s. The received data is thenoffline digital processed by a computer. The data is first resampled to56 Gsa/s and then processed by disclosed MMBE algorithms forpolarization demultiplexing, carrier frequency offset estimation andphase recovery before BER measurement.

FIG. 10 shows the measured back to back BER performance varying withOSNR for regular CMBE and disclosed MMBE schemes as a comparison (1000).It shows that, our scheme based on multi-modulus scheme MMBE (lowercurve) shows better BER performance compared with conventional CMBEscheme (upper curve). The constellations of X and Y pol. obtained by thetwo schemes at OSNR of 19.8 dB are also inserted in FIGS. 10 (1006 and1008). We can see an enhanced suppression of noise for the 9-pointconstellation obtained by our scheme.

We have disclosed and experimentally demonstrated a novel DSP scheme forQDB spectrum shaped PM-QPSK based on MMBE. The algorithms for this novelDSP scheme include CMMA for blind polarization de-multiplexing,multi-modulus QPSK partitioning FOE and two stages CPR with ML phaseestimation. The final signal is detected by MLSD for data BERmeasurement. The feasibility of the disclosed digital signal processingscheme is demonstrated by the experiment of 112 Gb/s PM-QPSK signalwhich is QDB shaped by a 25 GHz bandwidth waveshaper for NWDM channels.Our scheme shows better BER performance compared with conventional CMBEscheme.

FIG. 11 is a flow chart representation of a method 1100 for opticalcommunications. In some implementations, the method may be implementedat the receiver side of an optical communication system. At 1102, asignal comprising a quadrature duobinary modulated signal having a threemodulus constellation is received. In some implementations, the signalmay be produced by spectrally shaping I and Q symbols streams throughelectrical low-pass filters in the electrical domain. In someimplementations, the signal may be produced using an optical domainbandpass filter after I and Q symbols are modulated using optical domainQPSK. In some implementations, a constellation having a pre-determinednumber of moduli constellation (e.g., nine) may be used.

At 1104, channel equalization of the received signal is performed usinga constant multi-modulus to obtain a set of channel estimationcoefficients and a stream of symbols. In some implementations,techniques such as discussed with reference to FIG. 2 and FIG. 3 areused.

At 1106, based on modulus, the stream of symbols is partitioned intothree partitions. In some implementations, another predetermined numberof partitions are used (e.g., nine).

At 1108, carrier frequency offset of the signal is estimated, based onthe partitioned stream of symbols.

At 1110, a phase of the signal is recovered using a maximum likelihoodalgorithm.

At 1112, the partitioned stream of symbols is decoded to recover data.

FIG. 12 is a block diagram representation of an apparatus 12 forreceiving optical communications signals is depicted. The module 1202 isfor receiving a signal comprising a quadrature duobinary modulatedsignal having a three modulus constellation. The module 1204 is forperforming channel equalization of the received signal using a constantmulti-modulus to obtain a set of channel estimation coefficients and astream of symbols. The module 1206 is for partitioning, based onmodulus, the stream of symbols into three partitions. The module 1208 isfor estimating carrier frequency offset based on the partitioned streamof symbols. The module 1210 is for recovering a phase of the signalusing a maximum likelihood algorithm. The module 1212 is for decodingthe partitioned stream of symbols to recover data.

It will be appreciated that novel techniques for receiving modulatedoptical signals are disclosed. The disclosed techniques offer superiorperformance over conventional techniques by achieving an order ofmagnitude higher bit error rate (BER) for the same SNR, or alternativelyachieving 3 dB improvement for the same BER.

The disclosed and other embodiments, modules and the functionaloperations described in this document can be implemented in digitalelectronic circuitry, or in computer software, firmware, or hardware,including the structures disclosed in this document and their structuralequivalents, or in combinations of one or more of them. The disclosedand other embodiments can be implemented as one or more computer programproducts, i.e., one or more modules of computer program instructionsencoded on a computer readable medium for execution by, or to controlthe operation of, data processing apparatus. The computer readablemedium can be a machine-readable storage device, a machine-readablestorage substrate, a memory device, a composition of matter effecting amachine-readable propagated signal, or a combination of one or morethem. The term “data processing apparatus” encompasses all apparatus,devices, and machines for processing data, including by way of example aprogrammable processor, a computer, or multiple processors or computers.The apparatus can include, in addition to hardware, code that creates anexecution environment for the computer program in question, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them. A propagated signal is an artificially generated signal, e.g.,a machine-generated electrical, optical, or electromagnetic signal, thatis generated to encode information for transmission to suitable receiverapparatus.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a stand alone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., one or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network.

The processes and logic flows described in this document can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read only memory ora random access memory or both. The essential elements of a computer area processor for performing instructions and one or more memory devicesfor storing instructions and data. Generally, a computer will alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic, magneto optical disks, or optical disks. However, a computerneed not have such devices. Computer readable media suitable for storingcomputer program instructions and data include all forms of non volatilememory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto optical disks; and CD ROM and DVD-ROM disks. The processor andthe memory can be supplemented by, or incorporated in, special purposelogic circuitry.

While this patent document contains many specifics, these should not beconstrued as limitations on the scope of an invention that is claimed orof what may be claimed, but rather as descriptions of features specificto particular embodiments. Certain features that are described in thisdocument in the context of separate embodiments can also be implementedin combination in a single embodiment. Conversely, various features thatare described in the context of a single embodiment can also beimplemented in multiple embodiments separately or in any suitablesub-combination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asub-combination or a variation of a sub-combination. Similarly, whileoperations are depicted in the drawings in a particular order, thisshould not be understood as requiring that such operations be performedin the particular order shown or in sequential order, or that allillustrated operations be performed, to achieve desirable results.

Only a few examples and implementations are disclosed. Variations,modifications, and enhancements to the described examples andimplementations and other implementations can be made based on what isdisclosed.

What is claimed is what is disclosed and illustrated, including:
 1. Anoptical communications method, comprising: receiving a signal comprisinga quadrature duobinary modulated signal having a three modulusconstellation; performing channel equalization of the received signalusing a cascaded multi-modulus to obtain a set of channel estimationcoefficients and a stream of symbols; partitioning, based on modulus,the stream of symbols into three partitions, estimating carrierfrequency offset based on the partitioned stream of symbols; recoveringa phase of the signal using a maximum likelihood algorithm; decoding thepartitioned stream of symbols to recover data; and rotating at leastsome constellation points; wherein the rotating comprises rotating byπ/4 angle.
 2. The method recited in claim 1, wherein the rotatingoperation is performed during the operation of estimating the carrierfrequency offset.
 3. The method recited in claim 1, wherein the rotatingoperation is performed during the operation of recovering the phase ofthe signal.
 4. The method of claim 1, wherein the three modulusconstellation comprises a nine symbol quadrature amplitude modulationconstellation.
 5. An optical communications method, comprising:receiving a signal comprising a quadrature duobinary modulated signalhaving a three modulus constellation; performing channel equalization ofthe received signal using a cascaded multi-modulus to obtain a set ofchannel estimation coefficients and a stream of symbols; partitioning,based on modulus, the stream of symbols into three partitions,estimating carrier frequency offset based on the partitioned stream ofsymbols; recovering a phase of the signal using a maximum likelihoodalgorithm; decoding the partitioned stream of symbols to recover data;wherein the recovering operation is performed by first using aViterbi-Viterbi phase estimation algorithm and then using the maximumlikelihood algorithm.
 6. The method of claim 5, wherein the rotatingoperation is performed during the operation of estimating the carrierfrequency offset.
 7. The method of claim 5, wherein the rotatingoperation is performed during the operation of recovering the phase ofthe signal.
 8. The method of claim 5, wherein the three modulusconstellation comprises a nine symbol quadrature amplitude modulationconstellation.
 9. An optical communications apparatus, comprising: areceiver that receives a signal comprising a quadrature duobinarymodulated signal having a three modulus constellation; a channelequalizer that performs channel equalization of the received signalusing a cascaded multi-modulus to obtain a set of channel estimationcoefficients and a stream of symbols; a symbol partitioner thatpartitions, based on modulus, the stream of symbols into threepartitions; a frequency offset estimator that estimates carrierfrequency offset based on the partitioned stream of symbols; a phaserecoverer that recovers a phase of the signal using a maximum likelihoodalgorithm; and a data decoder that decodes the partitioned stream ofsymbols to recover data, wherein the phase recoverer recovers the phaseof the signal by first using a Viterbi-Viterbi phase estimationalgorithm and then using the maximum likelihood algorithm.
 10. Theapparatus recited in claim 9, further including: a rotator that rotatesat least some constellation points.
 11. The apparatus recited in claim10, wherein the frequency offset estimator includes the rotator.
 12. Theapparatus recited in claim 10, wherein the phase recoverer includes therotator.
 13. The apparatus recited in claim 9, wherein the three modulusconstellation comprises a nine symbol quadrature amplitude modulationconstellation.
 14. An optical communications method, comprising:receiving a signal comprising a quadrature duobinary modulated signalhaving a three modulus constellation; performing channel equalization ofthe received signal using a cascaded multi-modulus to obtain a set ofchannel estimation coefficients and a stream of symbols; partitioning,based on modulus, the stream of symbols into three partitions includingR1 symbols, R2 symbols and R3 symbols; rotating and then normalizing theR2 symbols; normalizing, without rotating, the R1 symbols; estimatingcarrier frequency offset by combining the rotated and normalized R2symbols together with the normalized R1 symbols; recovering a phase ofthe signal using a maximum likelihood algorithm; and decoding thepartitioned stream of symbols to recover data.
 15. The method of claim14, wherein the rotating and the normalizing are performed during theoperation of estimating the carrier frequency offset.
 16. The method ofclaim 14, wherein the rotating and the normalizing are performed duringthe operation of recovering the phase of the signal.
 17. The method ofclaim 14, wherein the three modulus constellation comprises a ninesymbol quadrature amplitude modulation constellation.
 18. An opticalcommunications apparatus, comprising: a receiver that receives a signalcomprising a quadrature duobinary modulated signal having a threemodulus constellation; a channel equalizer that performs channelequalization of the received signal using a cascaded multi-modulus toobtain a set of channel estimation coefficients and a stream of symbols;a symbol partitioner that partitions, based on modulus, the stream ofsymbols into three partitions including R1 symbols, R2 symbols and R3symbols; a rotator that rotates the R2 symbols; a normalizer thatnormalizes the rotated R2 symbols and the R1 symbols; a frequency offsetestimator that estimates carrier frequency offset by combining therotated and normalized R2 symbols together with the normalized R1symbols; a phase recoverer that recovers a phase of the signal using amaximum likelihood algorithm; and a data decoder that decodes thepartitioned stream of symbols to recover data.
 19. The apparatus ofclaim 18, wherein the frequency offset estimator includes the rotatorand the normalizer.
 20. The apparatus of claim 18, wherein the phaserecoverer includes the rotator and the normalizer.